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ARPN Journal of Engineering and Applied Sciences

2006-2015 Asian Research Publishing Network (ARPN). All rights reserved.

www.arpnjournals.com

1701

MULTIPLE CRACK DETECTION IN BEAMS FROM THE DIFFERENCES

IN CURVATURE MODE SHAPES

K. Ravi Prakash Babu1, B. Raghu Kumar2, K. L.Narayana2 and K. Mallikarjuna Rao11Deptartment of Mechanical Engineering, JNTUK Kakinada, India

2K L E F, K.L. University, Guntur, India

E-Mail: [email protected]

ABSTRACT

The presence of crack in a structure tends to modify its modal parameters (natural frequencies and mode shapes).The fact can be used inversely to predict the crack parameters (crack depth and its location) from measurement of the

changes in the modal parameters, once a functional relationship between them has been established. The machine

components like turbine blade can be treated as a cantilever beam and a shaft as a simply supported beam. Vibrationanalysis of cantilever beam and simply supported beam can be extended successfully to develop online crack detection

methodology in turbine blades and shafts. In the present work, finite element analysis of a cantilever and simply supported

beams for flexural vibrations has been considered by including two transverse open U-notches. The modal analysis has

been carried out on cantilever and simply supported beams with two U-notches and observed the influence of one U-notch

on the other for natural frequencies and mode shapes. This has been done by carrying out parametric studies using ANSYSsoftware to evaluate the natural frequencies and their corresponding mode shapes for different notch parameters (depths

and locations) of the cantilever and simply supported beams FEM model. Later, by using a central difference

approximation, curvature mode shapes were then calculated from the displacement mode shapes. The location and depthcorresponding to any peak on this curve becomes a possible notch location and depth. The identification procedure

presented in this study is believed to provide a useful tool for detection of medium size crack in a cantilever and simply

supported beam applications.

KEYWORDS:crack detection, vibration, FEM, displacement mode shapes, curvature mode shapes.

INTRODUCTION

Any localized crack in a structure reduces thestiffness in that area. These features are related to variation

in the dynamic properties, such as, decreases in natural

frequencies and variation of the modes of vibration of thestructure. One or more of above characteristics can be

used to detect and locate cracks. This property may beused to detect existence of crack or faults together with

location and its severity in a structural member. Rizos [1]

measured the amplitude at two points and proposed analgorithm to identify the location of crack. Pandey [2]

suggested a parameter, namely curvature of the deflected

shape of beam instead of change in frequencies to identify

the location of crack. Ostachowicz [3] proposed aprocedure for identification of a crack based on the

measurement of the deflection shape of the beam.

Ratcliffe [4] also developed a technique for identifying thelocation of structural damage in a beam using a 1D FEA.

A finite difference approximation called Laplacesdifferential operator was applied to the mode shapes to

identify the location of the damage. Wahab [5]investigated the application of the change in modal

curvatures to detect damage in a prestressed concrete

bridge. Lakshminarayana [6] carried out analytical work tostudy the effect of crack at different location and depth on

mode shape behaviour. Chandra Kishen [7] developed a

technique for damage detection using static test data.

Nahvi [8] established analytical as well as experimentalapproach to the crack detection in cantilever beams. Ravi

Prakash Babu [9] used differences in curvature mode

shapes to detect a crack in beams.

This paper deals with the technique and its

application of mode shapes to a cantilever and a simplysupported beam. The paper of Pandey et al. [2] shows a

quite interesting phenomenon: that is, the modal

curvatures are highly sensitive to crack and can be used tolocalize it. They used simulated data for a cantilever and a

simply supported beam model with single damage todemonstrate the applicability of the method. The cracked

beam was modeled by reducing the E-modulus of a certain

element. By plotting the difference in modal curvaturebetween the intact and the cracked beam, a peak appears at

the cracked element indicating the presence of a fault.

They used a central difference approximation to derive the

curvature mode shapes from the displacement modeshapes. An important remark could be observed from the

results of Pandey et al. [2]: that is, the difference in modal

curvature between the intact and the damaged beamshowed not only a high peak at the fault position but also

some small peaks at different undamaged locations for thehigher modes. To avoid this, a U-notch was modelled at

the location of faults instead of reducing the E-modulus ofa certain element. Also, analysis was carried out for beams

with two cracks instead of one.

So, this paper is done to study about the changesin mode shapes because of the presence of U-notches in

the cantilever and simply supported beams and is

concerned with investigation of the accuracy when using

the central difference approximation to compute the modalcurvature and determine the location of the U-notches and

to find out the reason of the presence of the misleading

small peaks. The application of this technique to

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constructions in which more than one fault positions existis investigated using a continuous beam with simulated

data. The results of this scenario will be analyzed in this

paper and U-notches will be detected and localized by

using the measured change in modal curvatures.So, as a summary, in the present work, a methodology for

predicting crack parameters (crack depth and its location)

in a cantilever and simply supported beam from changes incurvature mode shapes has been developed. Parametric

studies have been carried out using ANSYS Software to

evaluate mode shapes for different crack parameters

(depth and its location). Curvature mode shapes were then

calculated from the displacement mode shapes to identify

crack location and its severity in the cantilever and simplysupported beam.

PROBLEM FORMULATION

Figures-1(a) & 1(b) shows a cantilever and asimply supported beam of rectangular cross section, made

of mild steel with two U- notches. To find out mode

shapes associated with each natural frequency, FE analysishas been carried out using ANSYS Software for un-

notched and U-notched beam.

Figure-1(a). Cantilever beam with two U-notch cracks.

Figure-1(b). Simply supported beam with two U-notch cracks.

The mode shapes of the multiple U-notched

cantilever and simply supported beams are obtained for U-notches located at normalized distances (c/l & d/l) from

the fixed end of a cantilever beam and from left end

support of a simply supported beam with a normalizeddepth (a/h).

Figure-2. Discretised model of beam with two U-notches.

Figure-2 shows the discretised model (zoomed

near the position of U-notches) of a beam with two U-notches. Parametric studies have been carried out for thin

beam having length (l) = 260 mm, width (w) = 25 mm and

thickness (h) = 4.4 mm. The breadth (b) of each U-notchhas been kept as 0.32 mm. The U-notch locations from the

fixed end (c & d) of the cantilever beam have been taken

in different combinations near the fixed end, free end andmiddle of the beam. Similarly, U-notch locations from theleft end support (c & d) of the simply supported beam

have been taken in different combinations near the

supports and middle of the beam. The intensity of U-notch(a/h) was varied by increasing its depth over the range of

0.25 to 0.75 in the steps of 0.25. This represented the case

of a varying degree of crack at particular location. For

each model of the two U-notch locations, the first three

natural frequencies and corresponding mode shapes werecalculated using ANSYS software.

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Governing equation

Determination of modal parameters (natural

frequency & mode shape) in a beam is an Eigen value

problem. ANSYS Software is used for theoretical modalanalysis of the beam and the governing equation for

general Eigen value problem is:

0 XKXM (1)Disposing the brackets without ambiguity equation (1) is

rewritten as follows:

0KXXM (2)

Pre-multiplying both sides of equation (2) by1

M :

0111 MKXMXMM Now, XIXMM

1and

AXKXM 1

(say)

0AXXI (3)

Where, KMA1

= system matrix.

Assuming harmonic motion ;XX where, 2

equation (3) becomes

0 XIA (4)The characteristic of motion is then:

0 IA (5)

The n-rootsi

, where i=1, 2, 3n of the characteristic

equation (5) are called Eigen values.

The natural frequencies are found as:

ii , i = 1,2,3n. (6)

Substitution of the Eigen values (i

s) in (4) gives the

mode shapesi

X corresponding toi

. These are Eigen

vectors.

RESULTS AND DISCUSSION

In this analysis, it is assumed that crack is of U-

notched shape. The depth (a) and locations (c & d) ofthese notches are normalized to the height and length of

the cantilever and simply supported beams respectively.

The first three mode shapes for the beam were calculatedusing ANSYS software and were shown below for

different crack depths and crack location ratios.

Curvature finite difference approximationLocalized changes in stiffness result in a mode

shape that has a localized change in slope, therefore, this

feature will be studied as a possible parameter for crack

detection purposes. For a beam in bending the curvature(k) can be approximated by the second derivate of the

deflection:

(7)

In addition, numerical mode shape data is discretein space, thus the change in slope at each node can be

estimated using finite difference approximations. In thiswork, the central difference equation was used to

approximate the second derivate of the displacements ualong the x - direction at node i:

(8)

The term is the element length.

In this process meshing and node numbering is very

important. Equation (8) require the knowledge of thedisplacements at node i, node i -1 and node i+1 in order to

evaluate the curvature at node i. Thus, the value of the

curvature of the mode shapes could be calculated starting

from node 2 through node 261 in case of this beam. After

obtaining the curvature mode shapes the absolute

difference between the uncracked and cracked state isdetermined to improve crack detection.

(9)

As a result of this analysis, a set of curvature

vectors for different crack localizations are obtained.

Uncracked case

By using the same finite element model shown inFigure-2, linear mode shapes was performed in ANSYS.

The numerical results were exported to MATLAB to be

processed.The associated mode shapes were sketched

evaluating the displacements in y direction of the 261

equidistant nodes located at the bottom line of the beam.In order to unify the results from the different cases, mode

shapes were normalized by setting the largest grid point

displacement equal to 1. It can be noticed from figures thatall the mode shapes smoothed functions, what indicate the

absence of cracks. Cracked beam mode shapes will be

used to compare further results. Since changes in the

curvature are local in nature, they can be used to detect

and locate cracks in the beam.

Simple cracked case

The different combinations of crack locationscenarios are selected for studying the effect of localized

cracks in the beam. Although the reduction in natural

frequencies is related to the existence of crack and itsseverity, this feature cannot provide any usefulinformation about the location of the crack. Thus,

curvature mode shapes were calculated and compared with

the uncracked case. It can be seen that the maximumdifference value for each mode shape occurs in the crack

locations. In other areas of the beam this characteristic was

much smaller. Although the third mode shapes was the

most sensitive to the failure it is important that any of the

three curvature mode shapes peak at the cracked locations.

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Cantilever beam

Figure-3. Mode shapes for crack at c/l = 0.15 of depth a/h = 0.25 and crack at d/l = 0.8 of depth a/h = 0.75.

Figure-4. Difference curvature for crack at c/l = 0.15 of depth a/h = 0.25 and crack at d/l = 0.8 of depth a/h = 0.75.


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To examine the curvature mode shape techniquefor cantilever beam having several crack locations, the

span of the beam is discretized by 260 elements. Two

crack locations are assumed at a time. Firstly, two cracks

at c/l = 0.15 of depth a/h = 0.25 and at c/l = 0.8 of depth

a/h = 0.75 were considered. The first three displacementmode shapes are shown in Figure 3. The difference inmodal curvature between the uncracked and the cracked

beam is plotted in Figure 4 for the first three modes. For

mode 1 in Figure 4, it can be observed that the peak at c/l= 0.15 is very small comparing to that at d/l = 0.8. And

also, for mode 1 the modal curvature at c/l = 0.15 is less

and has very small values at the nodes next to it. Incontrast, at d/l = 0.8 high modal curvature takes place.

This indicates the severity of the crack depth ratio a/h =0.75 at d/l = 0.8 compared to crack depth ratio a/h = 0.25

at c/l = 0.15. And also, first mode shape modal curvature

is more sensitive near the fixed end compared to second

and third mode shape modal curvatures. Again second and

third mode shape modal curvatures are more sensitive nearthe free end compared to first mode shape modalcurvature. The same can be observed for some other crack

scenarios shown in Figures-5 & 6, 7 & 8, 9 & 10. So,

depending on the absolute ratio between the modalcurvature values for a particular mode at two different

locations, one peak can dominate the other. Therefore, one

can conclude that in case of several crack locations in astructure, all modes should be carefully examined.

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Simply supported beam




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To examine the curvature mode shape technique

for simply supported beam having several crack locations,

the span of the beam is also discretized by 260 elements.

Two crack locations are assumed at a time. Firstly, two

cracks at c/l = 0.15 of depth a/h = 0.25 and at c/l = 0.8 ofdepth a/h = 0.5 were considered. The first three

displacement mode shapes are shown in Figure 11. Thedifference in modal curvature between the uncracked and

the cracked beam is plotted in Figure 12 for the first three

modes. For all the modes in Figure 12, it can be observed

that the peak at c/l = 0.15 is very small comparing to thatat d/l = 0.8. In contrast, at d/l = 0.8 high modal curvature

takes place. This indicates the severity of the crack depth

ratio a/h = 0.5 at d/l = 0.8 compared to crack depth ratioa/h = 0.25 at c/l = 0.15. And also, third mode shape modal

curvature is more sensitive near the supports compared to

first and second mode shape modal curvatures. The same

can be observed in Figure 18 at d/l = 0.85. Again in Figure

14 third mode shape modal curvature is more sensitive

near the middle portion of the beam at d/l = 0.45 compared

to other mode shape modal curvature and the second mode

modal curvature is more sensitive near the one-fourth of

the length of the beam at c/l = 0.25. The same can be

observed for some other crack scenarios shown in Figure16 at d/l = 0.65 and in Figure 20 at c/l = 0.65 for second

mode shape. So, depending on the absolute ratio betweenthe modal curvature values for a particular mode at two

different locations, one peak can dominate the other.

Therefore, one can conclude that in case of several crack

locations in a structure, all modes should be carefullyexamined.

And also, the crack is assumed to affect stiffness

of the cantilever beam. The stiffness matrix of the crackedelement in the FEM model of the beam will replace the

stiffness matrix of the same element prior to damaging to

result in the global stiffness matrix. Thus frequencies and

mode shapes are obtained by solving the eigen value

problem [K] 2 [M] = 0. So it can be seen in Figure 3, 9,

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13 very clearly the changes in slopes and deviations inmode shape at crack location for crack depth ratio 0.75.

CONCLUSIONS

A method for identifying multiple crackparameters (crack depth and its location) in beams using

modal parameters has been attempted in the present paper.

Parametric studies have been carried out using ANSYSSoftware to evaluate modal parameters (natural

frequencies and mode shapes) for different crack

parameters. A theoretical study using simulated data for a

cantilever and simply supported beams has been

conducted. When more than one fault exists in thestructure, it is not possible to locate crack in all positions

from the results of only one mode. All modes should be

carefully examined in order to locate all existing faults.Also, due to the irregularities in the measured mode

shapes, a curve fitting can be applied by calculating the

curvature mode shapes using the central differenceapproximation. The curvature mode shapes technique forcrack localization in structures is investigated in this

paper. The results confirm that the application of the

curvature mode shape method to detect cracks inengineering structures seems to be promising. Techniques

for improving the quality of the measured mode shapes are

highly recommended. The identification procedure

presented in this paper is believed to provide a useful tool

for detection of medium size cracks in beams.

ACKNOWLEDGEMENTS

The author is grateful to the President, K L E FUniversity, for the kind permission to publish the paper. I

am also grateful to all the staff members of Mechanicaldepartment who have helped in carrying out the work.

REFERENCES

[1] P. F. Rizos., N Aspragathos. and A. D. Dimarogonas.

1990. Identification of crack location and magnitude

in a cantilever beam from the vibration modes,

Journal of Sound and Vibration. 138: 381-388.

[2] A. K. Pandey., M. Biswas and M. M. Samman. 1991.

Damage detection from change in curvature mode

shapes, Journal of Sound and Vibration. 145: 321-332.

[3]

W. M.Ostachowicz. and M. Krawczuk. 1991.Analysis of the effect of cracks on natural frequencies

of a cantilever beam, Journal of Sound and Vibration.

150: 191-201.

[4] C. P. Ratcliffe. 1997. Damage detection using a

modified Laplacian operator on mode shape data,

Journal of Sound and Vibration. 204(3): 505-517.

[5] M. Wahab and G Roeck. 1999. Damage detection inbridges using modal curvatures: application to a real

damage scenario. Journal of Sound and Vibration.226(2):217-235.

[6] K. Lakshminarayana. and C. Jebaraj. 1991. Sensitivity

analysis of local/global modal parameters foridentification of a crack in a beam. Journal of Sound

and Vibration. 228: 977-994.

[7] J. M. Chandra Kishen and Trisha Sain. 2004. Damage

detection using static test data, Journal of Structural

Engineering, India. 31(1): 15-21.

[8] H. Nahvi. and M. Jabbari. 2005. Crack detection inbeams using experimental modal data and finite

element model. International Journal of Mechanical

Sciences. 47: 1477-1497.

[9] K. Ravi Prakash Babu and G. Durga Prasad. 2012.

Crack detection in beams from the differences incurvature mode shapes. Journal of Structural

Engineering, India. 39(2): 237-242.

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